LMFDB - Embedded newform 690.2.e.a.551.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(551,690)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("690.551");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 12 x^{13} + 15 x^{12} - 4 x^{11} + 45 x^{10} - 66 x^{9} - 32 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + 3x^14 - 12x^13 + 15x^12 - 4x^11 + 45x^10 - 66x^9 - 32x^8 - 198x^7 + 405x^6 - 108x^5 + 1215x^4 - 2916x^3 + 2187x^2 - 4374*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			551.12
Root			$-1.62589 - 0.597052i$ of defining polynomial
Character	$\chi$	$=$	690.551
Dual form			690.2.e.a.551.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(-0.597052 + 1.62589i) q^{3} -1.00000 q^{4} -1.00000 q^{5} +(-1.62589 - 0.597052i) q^{6} +3.80421i q^{7} -1.00000i q^{8} +(-2.28706 - 1.94149i) q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +(-0.597052 + 1.62589i) q^{3} -1.00000 q^{4} -1.00000 q^{5} +(-1.62589 - 0.597052i) q^{6} +3.80421i q^{7} -1.00000i q^{8} +(-2.28706 - 1.94149i) q^{9} -1.00000i q^{10} -3.48413 q^{11} +(0.597052 - 1.62589i) q^{12} +3.19916 q^{13} -3.80421 q^{14} +(0.597052 - 1.62589i) q^{15} +1.00000 q^{16} -4.76053 q^{17} +(1.94149 - 2.28706i) q^{18} -1.88297i q^{19} +1.00000 q^{20} +(-6.18523 - 2.27131i) q^{21} -3.48413i q^{22} +(0.785398 + 4.73108i) q^{23} +(1.62589 + 0.597052i) q^{24} +1.00000 q^{25} +3.19916i q^{26} +(4.52214 - 2.55934i) q^{27} -3.80421i q^{28} -7.50582i q^{29} +(1.62589 + 0.597052i) q^{30} -3.72122 q^{31} +1.00000i q^{32} +(2.08021 - 5.66482i) q^{33} -4.76053i q^{34} -3.80421i q^{35} +(2.28706 + 1.94149i) q^{36} -4.88527i q^{37} +1.88297 q^{38} +(-1.91007 + 5.20149i) q^{39} +1.00000i q^{40} +3.99062i q^{41} +(2.27131 - 6.18523i) q^{42} +6.96125i q^{43} +3.48413 q^{44} +(2.28706 + 1.94149i) q^{45} +(-4.73108 + 0.785398i) q^{46} -10.8399i q^{47} +(-0.597052 + 1.62589i) q^{48} -7.47199 q^{49} +1.00000i q^{50} +(2.84229 - 7.74011i) q^{51} -3.19916 q^{52} -7.07396 q^{53} +(2.55934 + 4.52214i) q^{54} +3.48413 q^{55} +3.80421 q^{56} +(3.06151 + 1.12423i) q^{57} +7.50582 q^{58} +7.35646i q^{59} +(-0.597052 + 1.62589i) q^{60} -3.31936i q^{61} -3.72122i q^{62} +(7.38582 - 8.70044i) q^{63} -1.00000 q^{64} -3.19916 q^{65} +(5.66482 + 2.08021i) q^{66} +3.70440i q^{67} +4.76053 q^{68} +(-8.16116 - 1.54773i) q^{69} +3.80421 q^{70} +9.92565i q^{71} +(-1.94149 + 2.28706i) q^{72} +3.18473 q^{73} +4.88527 q^{74} +(-0.597052 + 1.62589i) q^{75} +1.88297i q^{76} -13.2543i q^{77} +(-5.20149 - 1.91007i) q^{78} -10.1264i q^{79} -1.00000 q^{80} +(1.46126 + 8.88058i) q^{81} -3.99062 q^{82} -12.8088 q^{83} +(6.18523 + 2.27131i) q^{84} +4.76053 q^{85} -6.96125 q^{86} +(12.2037 + 4.48137i) q^{87} +3.48413i q^{88} -11.0510 q^{89} +(-1.94149 + 2.28706i) q^{90} +12.1703i q^{91} +(-0.785398 - 4.73108i) q^{92} +(2.22176 - 6.05031i) q^{93} +10.8399 q^{94} +1.88297i q^{95} +(-1.62589 - 0.597052i) q^{96} -2.98155i q^{97} -7.47199i q^{98} +(7.96840 + 6.76439i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 16 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) $ 16 * q - 16 * q^4 - 16 * q^5 + 2 * q^6 + 2 * q^9	$ 16 q - 16 q^{4} - 16 q^{5} + 2 q^{6} + 2 q^{9} - 12 q^{11} - 12 q^{14} + 16 q^{16} + 8 q^{18} + 16 q^{20} - 4 q^{21} + 4 q^{23} - 2 q^{24} + 16 q^{25} + 24 q^{27} - 2 q^{30} + 4 q^{31} - 28 q^{33} - 2 q^{36} - 16 q^{38} - 8 q^{39} + 12 q^{44} - 2 q^{45} - 4 q^{46} - 4 q^{49} - 2 q^{51} - 8 q^{53} - 26 q^{54} + 12 q^{55} + 12 q^{56} + 28 q^{57} - 8 q^{58} - 16 q^{64} + 10 q^{66} + 30 q^{69} + 12 q^{70} - 8 q^{72} - 16 q^{73} - 24 q^{74} - 12 q^{78} - 16 q^{80} + 22 q^{81} - 16 q^{82} - 40 q^{83} + 4 q^{84} - 40 q^{86} + 20 q^{87} + 80 q^{89} - 8 q^{90} - 4 q^{92} - 4 q^{93} - 24 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100}) $ 16 * q - 16 * q^4 - 16 * q^5 + 2 * q^6 + 2 * q^9 - 12 * q^11 - 12 * q^14 + 16 * q^16 + 8 * q^18 + 16 * q^20 - 4 * q^21 + 4 * q^23 - 2 * q^24 + 16 * q^25 + 24 * q^27 - 2 * q^30 + 4 * q^31 - 28 * q^33 - 2 * q^36 - 16 * q^38 - 8 * q^39 + 12 * q^44 - 2 * q^45 - 4 * q^46 - 4 * q^49 - 2 * q^51 - 8 * q^53 - 26 * q^54 + 12 * q^55 + 12 * q^56 + 28 * q^57 - 8 * q^58 - 16 * q^64 + 10 * q^66 + 30 * q^69 + 12 * q^70 - 8 * q^72 - 16 * q^73 - 24 * q^74 - 12 * q^78 - 16 * q^80 + 22 * q^81 - 16 * q^82 - 40 * q^83 + 4 * q^84 - 40 * q^86 + 20 * q^87 + 80 * q^89 - 8 * q^90 - 4 * q^92 - 4 * q^93 - 24 * q^94 + 2 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−0.597052	+	1.62589i	−0.344708	+	0.938710i
$3$	−0.597052	+	1.62589i	−0.344708	+	0.938710i
$4$	−1.00000			−0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$	−1.62589	−	0.597052i	−0.663768	−	0.243746i
$7$			3.80421i			1.43786i	0.695085	+	0.718928i	$0.255367\pi$
$7$			3.80421i			1.43786i	−0.695085	+	0.718928i	$0.744633\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−2.28706	−	1.94149i	−0.762352	−	0.647162i
$10$		−	1.00000i		−	0.316228i
$11$	−3.48413			−1.05050			−0.525252	−	0.850947i	$-0.676029\pi$
$11$	−3.48413			−1.05050			−0.525252	+	0.850947i	$0.676029\pi$
$12$	0.597052	−	1.62589i	0.172354	−	0.469355i
$13$	3.19916			0.887287			0.443644	−	0.896203i	$-0.353686\pi$
$13$	3.19916			0.887287			0.443644	+	0.896203i	$0.353686\pi$
$14$	−3.80421			−1.01672
$15$	0.597052	−	1.62589i	0.154158	−	0.419804i
$16$	1.00000			0.250000
$17$	−4.76053			−1.15460			−0.577299	−	0.816533i	$-0.695893\pi$
$17$	−4.76053			−1.15460			−0.577299	+	0.816533i	$0.695893\pi$
$18$	1.94149	−	2.28706i	0.457613	−	0.539064i
$19$		−	1.88297i		−	0.431984i	−0.976395	−	0.215992i	$-0.930702\pi$
$19$		−	1.88297i		−	0.431984i	0.976395	−	0.215992i	$-0.0692985\pi$
$20$	1.00000			0.223607
$21$	−6.18523	−	2.27131i	−1.34973	−	0.495641i
$22$		−	3.48413i		−	0.742818i
$23$	0.785398	+	4.73108i	0.163767	+	0.986499i
$23$	0.785398	+	4.73108i	0.163767	+	0.986499i
$24$	1.62589	+	0.597052i	0.331884	+	0.121873i
$25$	1.00000			0.200000
$26$			3.19916i			0.627407i
$27$	4.52214	−	2.55934i	0.870287	−	0.492545i
$28$		−	3.80421i		−	0.718928i
$29$		−	7.50582i		−	1.39380i	−0.717170	−	0.696898i	$-0.754563\pi$
$29$		−	7.50582i		−	1.39380i	0.717170	−	0.696898i	$-0.245437\pi$
$30$	1.62589	+	0.597052i	0.296846	+	0.109006i
$31$	−3.72122			−0.668351			−0.334176	−	0.942511i	$-0.608458\pi$
$31$	−3.72122			−0.668351			−0.334176	+	0.942511i	$0.608458\pi$
$32$			1.00000i			0.176777i
$33$	2.08021	−	5.66482i	0.362117	−	0.986118i
$34$		−	4.76053i		−	0.816424i
$35$		−	3.80421i		−	0.643028i
$36$	2.28706	+	1.94149i	0.381176	+	0.323581i
$37$		−	4.88527i		−	0.803133i	−0.915830	−	0.401567i	$-0.868466\pi$
$37$		−	4.88527i		−	0.803133i	0.915830	−	0.401567i	$-0.131534\pi$
$38$	1.88297			0.305459
$39$	−1.91007	+	5.20149i	−0.305855	+	0.832905i
$40$			1.00000i			0.158114i
$41$			3.99062i			0.623231i	0.950208	+	0.311615i	$0.100870\pi$
$41$			3.99062i			0.623231i	−0.950208	+	0.311615i	$0.899130\pi$
$42$	2.27131	−	6.18523i	0.350471	−	0.954402i
$43$			6.96125i			1.06158i	0.847503	+	0.530791i	$0.178105\pi$
$43$			6.96125i			1.06158i	−0.847503	+	0.530791i	$0.821895\pi$
$44$	3.48413			0.525252
$45$	2.28706	+	1.94149i	0.340934	+	0.289420i
$46$	−4.73108	+	0.785398i	−0.697560	+	0.115801i
$47$		−	10.8399i		−	1.58116i	−0.612357	−	0.790581i	$-0.709779\pi$
$47$		−	10.8399i		−	1.58116i	0.612357	−	0.790581i	$-0.290221\pi$
$48$	−0.597052	+	1.62589i	−0.0861771	+	0.234677i
$49$	−7.47199			−1.06743
$50$			1.00000i			0.141421i
$51$	2.84229	−	7.74011i	0.398000	−	1.08383i
$52$	−3.19916			−0.443644
$53$	−7.07396			−0.971683			−0.485841	−	0.874047i	$-0.661487\pi$
$53$	−7.07396			−0.971683			−0.485841	+	0.874047i	$0.661487\pi$
$54$	2.55934	+	4.52214i	0.348282	+	0.615386i
$55$	3.48413			0.469800
$56$	3.80421			0.508359
$57$	3.06151	+	1.12423i	0.405507	+	0.148908i
$58$	7.50582			0.985562
$59$			7.35646i			0.957730i	0.877889	+	0.478865i	$0.158952\pi$
$59$			7.35646i			0.957730i	−0.877889	+	0.478865i	$0.841048\pi$
$60$	−0.597052	+	1.62589i	−0.0770791	+	0.209902i
$61$		−	3.31936i		−	0.425000i	−0.977161	−	0.212500i	$-0.931839\pi$
$61$		−	3.31936i		−	0.425000i	0.977161	−	0.212500i	$-0.0681606\pi$
$62$		−	3.72122i		−	0.472596i
$63$	7.38582	−	8.70044i	0.930526	−	1.09615i
$64$	−1.00000			−0.125000
$65$	−3.19916			−0.396807
$66$	5.66482	+	2.08021i	0.697291	+	0.256056i
$67$			3.70440i			0.452564i	0.974062	+	0.226282i	$0.0726571\pi$
$67$			3.70440i			0.452564i	−0.974062	+	0.226282i	$0.927343\pi$
$68$	4.76053			0.577299
$69$	−8.16116	−	1.54773i	−0.982488	−	0.186325i
$70$	3.80421			0.454690
$71$			9.92565i			1.17796i	0.808148	+	0.588979i	$0.200470\pi$
$71$			9.92565i			1.17796i	−0.808148	+	0.588979i	$0.799530\pi$
$72$	−1.94149	+	2.28706i	−0.228806	+	0.269532i
$73$	3.18473			0.372744			0.186372	−	0.982479i	$-0.440327\pi$
$73$	3.18473			0.372744			0.186372	+	0.982479i	$0.440327\pi$
$74$	4.88527			0.567901
$75$	−0.597052	+	1.62589i	−0.0689417	+	0.187742i
$76$			1.88297i			0.215992i
$77$		−	13.2543i		−	1.51047i
$78$	−5.20149	−	1.91007i	−0.588953	−	0.216272i
$79$		−	10.1264i		−	1.13931i	−0.821885	−	0.569653i	$-0.807078\pi$
$79$		−	10.1264i		−	1.13931i	0.821885	−	0.569653i	$-0.192922\pi$
$80$	−1.00000			−0.111803
$81$	1.46126	+	8.88058i	0.162362	+	0.986731i
$82$	−3.99062			−0.440691
$83$	−12.8088			−1.40595			−0.702974	−	0.711215i	$-0.748145\pi$
$83$	−12.8088			−1.40595			−0.702974	+	0.711215i	$0.748145\pi$
$84$	6.18523	+	2.27131i	0.674864	+	0.247820i
$85$	4.76053			0.516352
$86$	−6.96125			−0.750651
$87$	12.2037	+	4.48137i	1.30837	+	0.480453i
$88$			3.48413i			0.371409i
$89$	−11.0510			−1.17140			−0.585700	−	0.810528i	$-0.699180\pi$
$89$	−11.0510			−1.17140			−0.585700	+	0.810528i	$0.699180\pi$
$90$	−1.94149	+	2.28706i	−0.204651	+	0.241077i
$91$			12.1703i			1.27579i
$92$	−0.785398	−	4.73108i	−0.0818834	−	0.493250i
$93$	2.22176	−	6.05031i	0.230386	−	0.627388i
$94$	10.8399			1.11805
$95$			1.88297i			0.193189i
$96$	−1.62589	−	0.597052i	−0.165942	−	0.0609364i
$97$		−	2.98155i		−	0.302730i	−0.988478	−	0.151365i	$-0.951633\pi$
$97$		−	2.98155i		−	0.302730i	0.988478	−	0.151365i	$-0.0483669\pi$
$98$		−	7.47199i		−	0.754785i
$99$	7.96840	+	6.76439i	0.800854	+	0.679846i
$100$	−1.00000			−0.100000
$101$			9.15523i			0.910979i	0.890241	+	0.455490i	$0.150536\pi$
$101$			9.15523i			0.910979i	−0.890241	+	0.455490i	$0.849464\pi$
$102$	7.74011	+	2.84229i	0.766385	+	0.281428i
$103$			16.0594i			1.58238i	0.611570	+	0.791190i	$0.290538\pi$
$103$			16.0594i			1.58238i	−0.611570	+	0.791190i	$0.709462\pi$
$104$		−	3.19916i		−	0.313703i
$105$	6.18523	+	2.27131i	0.603617	+	0.221657i
$106$		−	7.07396i		−	0.687083i
$107$	17.6195			1.70334			0.851669	−	0.524080i	$-0.175591\pi$
$107$	17.6195			1.70334			0.851669	+	0.524080i	$0.175591\pi$
$108$	−4.52214	+	2.55934i	−0.435143	+	0.246273i
$109$			12.7316i			1.21947i	0.792607	+	0.609733i	$0.208723\pi$
$109$			12.7316i			1.21947i	−0.792607	+	0.609733i	$0.791277\pi$
$110$			3.48413i			0.332198i
$111$	7.94293	+	2.91676i	0.753909	+	0.276847i
$112$			3.80421i			0.359464i
$113$	−9.34208			−0.878829			−0.439414	−	0.898284i	$-0.644814\pi$
$113$	−9.34208			−0.878829			−0.439414	+	0.898284i	$0.644814\pi$
$114$	−1.12423	+	3.06151i	−0.105294	+	0.286737i
$115$	−0.785398	−	4.73108i	−0.0732387	−	0.441176i
$116$			7.50582i			0.696898i
$117$	−7.31666	−	6.21113i	−0.676426	−	0.574219i
$118$	−7.35646			−0.677217
$119$		−	18.1100i		−	1.66014i
$120$	−1.62589	−	0.597052i	−0.148423	−	0.0545032i
$121$	1.13914			0.103558
$122$	3.31936			0.300520
$123$	−6.48833	−	2.38261i	−0.585033	−	0.214833i
$124$	3.72122			0.334176
$125$	−1.00000			−0.0894427
$126$	8.70044	+	7.38582i	0.775097	+	0.657981i
$127$	−15.6846			−1.39178			−0.695892	−	0.718146i	$-0.744991\pi$
$127$	−15.6846			−1.39178			−0.695892	+	0.718146i	$0.744991\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−11.3183	−	4.15623i	−0.996517	−	0.365936i
$130$		−	3.19916i		−	0.280585i
$131$			10.5503i			0.921787i	0.887455	+	0.460894i	$0.152471\pi$
$131$			10.5503i			0.921787i	−0.887455	+	0.460894i	$0.847529\pi$
$132$	−2.08021	+	5.66482i	−0.181059	+	0.493059i
$133$	7.16322			0.621130
$134$	−3.70440			−0.320011
$135$	−4.52214	+	2.55934i	−0.389204	+	0.220273i
$136$			4.76053i			0.408212i
$137$	−16.2614			−1.38930			−0.694652	−	0.719346i	$-0.744442\pi$
$137$	−16.2614			−1.38930			−0.694652	+	0.719346i	$0.744442\pi$
$138$	1.54773	−	8.16116i	0.131752	−	0.694724i
$139$	−23.3435			−1.97997			−0.989984	−	0.141180i	$-0.954910\pi$
$139$	−23.3435			−1.97997			−0.989984	+	0.141180i	$0.954910\pi$
$140$			3.80421i			0.321514i
$141$	17.6245	+	6.47199i	1.48425	+	0.545040i
$142$	−9.92565			−0.832942
$143$	−11.1463			−0.932099
$144$	−2.28706	−	1.94149i	−0.190588	−	0.161791i
$145$			7.50582i			0.623324i
$146$			3.18473i			0.263570i
$147$	4.46117	−	12.1487i	0.367951	−	1.00200i
$148$			4.88527i			0.401567i
$149$	7.01303			0.574530			0.287265	−	0.957851i	$-0.407254\pi$
$149$	7.01303			0.574530			0.287265	+	0.957851i	$0.407254\pi$
$150$	−1.62589	−	0.597052i	−0.132754	−	0.0487491i
$151$	−14.3153			−1.16496			−0.582481	−	0.812844i	$-0.697918\pi$
$151$	−14.3153			−1.16496			−0.582481	+	0.812844i	$0.697918\pi$
$152$	−1.88297			−0.152729
$153$	10.8876	+	9.24251i	0.880210	+	0.747212i
$154$	13.2543			1.06807
$155$	3.72122			0.298896
$156$	1.91007	−	5.20149i	0.152928	−	0.416453i
$157$			16.3884i			1.30794i	0.756521	+	0.653970i	$0.226898\pi$
$157$			16.3884i			1.30794i	−0.756521	+	0.653970i	$0.773102\pi$
$158$	10.1264			0.805611
$159$	4.22352	−	11.5015i	0.334947	−	0.912128i
$160$		−	1.00000i		−	0.0790569i
$161$	−17.9980	+	2.98781i	−1.41844	+	0.235473i
$162$	−8.88058	+	1.46126i	−0.697724	+	0.114807i
$163$	5.26525			0.412407			0.206203	−	0.978509i	$-0.433889\pi$
$163$	5.26525			0.412407			0.206203	+	0.978509i	$0.433889\pi$
$164$		−	3.99062i		−	0.311615i
$165$	−2.08021	+	5.66482i	−0.161944	+	0.441006i
$166$		−	12.8088i		−	0.994156i
$167$			7.87574i			0.609443i	0.952441	+	0.304722i	$0.0985634\pi$
$167$			7.87574i			0.609443i	−0.952441	+	0.304722i	$0.901437\pi$
$168$	−2.27131	+	6.18523i	−0.175235	+	0.477201i
$169$	−2.76537			−0.212721
$170$			4.76053i			0.365116i
$171$	−3.65577	+	4.30647i	−0.279564	+	0.329324i
$172$		−	6.96125i		−	0.530791i
$173$		−	2.12595i		−	0.161633i	−0.996729	−	0.0808165i	$-0.974247\pi$
$173$		−	2.12595i		−	0.161633i	0.996729	−	0.0808165i	$-0.0257528\pi$
$174$	−4.48137	+	12.2037i	−0.339732	+	0.925157i
$175$			3.80421i			0.287571i
$176$	−3.48413			−0.262626
$177$	−11.9608	−	4.39219i	−0.899031	−	0.330138i
$178$		−	11.0510i		−	0.828305i
$179$		−	12.4614i		−	0.931409i	−0.884940	−	0.465704i	$-0.845801\pi$
$179$		−	12.4614i		−	0.931409i	0.884940	−	0.465704i	$-0.154199\pi$
$180$	−2.28706	−	1.94149i	−0.170467	−	0.144710i
$181$			10.2922i			0.765012i	0.923953	+	0.382506i	$0.124939\pi$
$181$			10.2922i			0.765012i	−0.923953	+	0.382506i	$0.875061\pi$
$182$	−12.1703			−0.902120
$183$	5.39692	+	1.98183i	0.398952	+	0.146501i
$184$	4.73108	−	0.785398i	0.348780	−	0.0579003i
$185$			4.88527i			0.359172i
$186$	6.05031	+	2.22176i	0.443630	+	0.162908i
$187$	16.5863			1.21291
$188$			10.8399i			0.790581i
$189$	9.73626	+	17.2032i	0.708209	+	1.25135i
$190$	−1.88297			−0.136605
$191$	22.5181			1.62935			0.814676	−	0.579916i	$-0.196915\pi$
$191$	22.5181			1.62935			0.814676	+	0.579916i	$0.196915\pi$
$192$	0.597052	−	1.62589i	0.0430885	−	0.117339i
$193$	12.3106			0.886134			0.443067	−	0.896489i	$-0.353890\pi$
$193$	12.3106			0.886134			0.443067	+	0.896489i	$0.353890\pi$
$194$	2.98155			0.214063
$195$	1.91007	−	5.20149i	0.136783	−	0.372487i
$196$	7.47199			0.533714
$197$			0.321151i			0.0228811i	0.999935	+	0.0114405i	$0.00364171\pi$
$197$			0.321151i			0.0228811i	−0.999935	+	0.0114405i	$0.996358\pi$
$198$	−6.76439	+	7.96840i	−0.480724	+	0.566289i
$199$		−	14.6664i		−	1.03968i	−0.854265	−	0.519838i	$-0.825992\pi$
$199$		−	14.6664i		−	1.03968i	0.854265	−	0.519838i	$-0.174008\pi$
$200$		−	1.00000i		−	0.0707107i
$201$	−6.02295	−	2.21172i	−0.424826	−	0.156003i
$202$	−9.15523			−0.644160
$203$	28.5537			2.00408
$204$	−2.84229	+	7.74011i	−0.199000	+	0.541916i
$205$		−	3.99062i		−	0.278717i
$206$	−16.0594			−1.11891
$207$	7.38909	−	12.3451i	0.513577	−	0.858043i
$208$	3.19916			0.221822
$209$			6.56052i			0.453801i
$210$	−2.27131	+	6.18523i	−0.156735	+	0.426822i
$211$	12.3848			0.852604			0.426302	−	0.904581i	$-0.359816\pi$
$211$	12.3848			0.852604			0.426302	+	0.904581i	$0.359816\pi$
$212$	7.07396			0.485841
$213$	−16.1380	−	5.92613i	−1.10576	−	0.406052i
$214$			17.6195i			1.20444i
$215$		−	6.96125i		−	0.474753i
$216$	−2.55934	−	4.52214i	−0.174141	−	0.307693i
$217$		−	14.1563i		−	0.960992i
$218$	−12.7316			−0.862292
$219$	−1.90145	+	5.17803i	−0.128488	+	0.349899i
$220$	−3.48413			−0.234900
$221$	−15.2297			−1.02446
$222$	−2.91676	+	7.94293i	−0.195760	+	0.533094i
$223$	5.19193			0.347677			0.173839	−	0.984774i	$-0.444383\pi$
$223$	5.19193			0.347677			0.173839	+	0.984774i	$0.444383\pi$
$224$	−3.80421			−0.254179
$225$	−2.28706	−	1.94149i	−0.152470	−	0.129432i
$226$		−	9.34208i		−	0.621426i
$227$	16.7298			1.11040			0.555199	−	0.831718i	$-0.312642\pi$
$227$	16.7298			1.11040			0.555199	+	0.831718i	$0.312642\pi$
$228$	−3.06151	−	1.12423i	−0.202754	−	0.0744542i
$229$			18.2985i			1.20920i	0.796530	+	0.604599i	$0.206666\pi$
$229$			18.2985i			1.20920i	−0.796530	+	0.604599i	$0.793334\pi$
$230$	4.73108	−	0.785398i	0.311958	−	0.0517876i
$231$	21.5501	+	7.91354i	1.41790	+	0.520672i
$232$	−7.50582			−0.492781
$233$			6.92267i			0.453519i	0.973951	+	0.226759i	$0.0728131\pi$
$233$			6.92267i			0.453519i	−0.973951	+	0.226759i	$0.927187\pi$
$234$	6.21113	−	7.31666i	0.406034	−	0.478305i
$235$			10.8399i			0.707117i
$236$		−	7.35646i		−	0.478865i
$237$	16.4644	+	6.04597i	1.06948	+	0.392728i
$238$	18.1100			1.17390
$239$			2.10045i			0.135867i	0.997690	+	0.0679335i	$0.0216406\pi$
$239$			2.10045i			0.135867i	−0.997690	+	0.0679335i	$0.978359\pi$
$240$	0.597052	−	1.62589i	0.0385396	−	0.104951i
$241$		−	0.0258170i		−	0.00166302i	−1.00000		0.000831511i	$-0.999735\pi$
$241$		−	0.0258170i		−	0.00166302i	1.00000		0.000831511i	$-0.000264678\pi$
$242$			1.13914i			0.0732268i
$243$	−15.3113	−	2.92632i	−0.982222	−	0.187724i
$244$			3.31936i			0.212500i
$245$	7.47199			0.477368
$246$	2.38261	−	6.48833i	0.151910	−	0.413681i
$247$		−	6.02393i		−	0.383294i
$248$			3.72122i			0.236298i
$249$	7.64752	−	20.8257i	0.484642	−	1.31978i
$250$		−	1.00000i		−	0.0632456i
$251$	7.11730			0.449240			0.224620	−	0.974446i	$-0.427886\pi$
$251$	7.11730			0.449240			0.224620	+	0.974446i	$0.427886\pi$
$252$	−7.38582	+	8.70044i	−0.465263	+	0.548076i
$253$	−2.73642	−	16.4837i	−0.172038	−	1.03632i
$254$		−	15.6846i		−	0.984141i
$255$	−2.84229	+	7.74011i	−0.177991	+	0.484705i
$256$	1.00000			0.0625000
$257$			5.27667i			0.329150i	0.986365	+	0.164575i	$0.0526252\pi$
$257$			5.27667i			0.329150i	−0.986365	+	0.164575i	$0.947375\pi$
$258$	4.15623	−	11.3183i	0.258756	−	0.704644i
$259$	18.5846			1.15479
$260$	3.19916			0.198404
$261$	−14.5725	+	17.1662i	−0.902012	+	1.06256i
$262$	−10.5503			−0.651802
$263$	13.6544			0.841967			0.420984	−	0.907068i	$-0.361685\pi$
$263$	13.6544			0.841967			0.420984	+	0.907068i	$0.361685\pi$
$264$	−5.66482	−	2.08021i	−0.348645	−	0.128028i
$265$	7.07396			0.434550
$266$			7.16322i			0.439205i
$267$	6.59801	−	17.9677i	0.403792	−	1.09961i
$268$		−	3.70440i		−	0.226282i
$269$		−	14.9128i		−	0.909249i	−0.890683	−	0.454624i	$-0.849774\pi$
$269$		−	14.9128i		−	0.909249i	0.890683	−	0.454624i	$-0.150226\pi$
$270$	−2.55934	−	4.52214i	−0.155757	−	0.275209i
$271$	−22.4734			−1.36516			−0.682580	−	0.730811i	$-0.739142\pi$
$271$	−22.4734			−1.36516			−0.682580	+	0.730811i	$0.739142\pi$
$272$	−4.76053			−0.288650
$273$	−19.7876	−	7.26629i	−1.19760	−	0.439776i
$274$		−	16.2614i		−	0.982387i
$275$	−3.48413			−0.210101
$276$	8.16116	+	1.54773i	0.491244	+	0.0931625i
$277$	−25.7920			−1.54969			−0.774846	−	0.632150i	$-0.782173\pi$
$277$	−25.7920			−1.54969			−0.774846	+	0.632150i	$0.782173\pi$
$278$		−	23.3435i		−	1.40005i
$279$	8.51065	+	7.22470i	0.509519	+	0.432532i
$280$	−3.80421			−0.227345
$281$	5.19176			0.309714			0.154857	−	0.987937i	$-0.450508\pi$
$281$	5.19176			0.309714			0.154857	+	0.987937i	$0.450508\pi$
$282$	−6.47199	+	17.6245i	−0.385401	+	1.04953i
$283$		−	6.60335i		−	0.392528i	−0.980551	−	0.196264i	$-0.937119\pi$
$283$		−	6.60335i		−	0.392528i	0.980551	−	0.196264i	$-0.0628810\pi$
$284$		−	9.92565i		−	0.588979i
$285$	−3.06151	−	1.12423i	−0.181348	−	0.0665939i
$286$		−	11.1463i		−	0.659093i
$287$	−15.1812			−0.896115
$288$	1.94149	−	2.28706i	0.114403	−	0.134766i
$289$	5.66264			0.333097
$290$	−7.50582			−0.440757
$291$	4.84768	+	1.78014i	0.284176	+	0.104354i
$292$	−3.18473			−0.186372
$293$	30.6756			1.79209			0.896044	−	0.443964i	$-0.146428\pi$
$293$	30.6756			1.79209			0.896044	+	0.443964i	$0.146428\pi$
$294$	12.1487	+	4.46117i	0.708524	+	0.260181i
$295$		−	7.35646i		−	0.428310i
$296$	−4.88527			−0.283951
$297$	−15.7557	+	8.91707i	−0.914240	+	0.517421i
$298$			7.01303i			0.406254i
$299$	2.51261	+	15.1355i	0.145308	+	0.875308i
$300$	0.597052	−	1.62589i	0.0344708	−	0.0938710i
$301$	−26.4820			−1.52640
$302$		−	14.3153i		−	0.823753i
$303$	−14.8854	−	5.46615i	−0.855145	−	0.314022i
$304$		−	1.88297i		−	0.107996i
$305$			3.31936i			0.190066i
$306$	−9.24251	+	10.8876i	−0.528359	+	0.622403i
$307$	19.3975			1.10707			0.553537	−	0.832825i	$-0.313278\pi$
$307$	19.3975			1.10707			0.553537	+	0.832825i	$0.313278\pi$
$308$			13.2543i			0.755236i
$309$	−26.1109	−	9.58831i	−1.48540	−	0.545460i
$310$			3.72122i			0.211351i
$311$		−	16.8790i		−	0.957118i	−0.878055	−	0.478559i	$-0.841159\pi$
$311$		−	16.8790i		−	0.957118i	0.878055	−	0.478559i	$-0.158841\pi$
$312$	5.20149	+	1.91007i	0.294477	+	0.108136i
$313$		−	26.7256i		−	1.51062i	−0.655367	−	0.755310i	$-0.727486\pi$
$313$		−	26.7256i		−	1.51062i	0.655367	−	0.755310i	$-0.272514\pi$
$314$	−16.3884			−0.924853
$315$	−7.38582	+	8.70044i	−0.416144	+	0.490214i
$316$			10.1264i			0.569653i
$317$			7.55156i			0.424138i	0.977255	+	0.212069i	$0.0680201\pi$
$317$			7.55156i			0.424138i	−0.977255	+	0.212069i	$0.931980\pi$
$318$	11.5015	+	4.22352i	0.644972	+	0.236843i
$319$			26.1512i			1.46419i
$320$	1.00000			0.0559017
$321$	−10.5197	+	28.6474i	−0.587155	+	1.59894i
$322$	−2.98781	−	17.9980i	−0.166504	−	1.00299i
$323$			8.96395i			0.498768i
$324$	−1.46126	−	8.88058i	−0.0811810	−	0.493366i
$325$	3.19916			0.177457
$326$			5.26525i			0.291615i
$327$	−20.7002	−	7.60143i	−1.14472	−	0.420360i
$328$	3.99062			0.220345
$329$	41.2372			2.27348
$330$	−5.66482	−	2.08021i	−0.311838	−	0.114512i
$331$	25.3475			1.39323			0.696613	−	0.717447i	$-0.254690\pi$
$331$	25.3475			1.39323			0.696613	+	0.717447i	$0.254690\pi$
$332$	12.8088			0.702974
$333$	−9.48469	+	11.1729i	−0.519758	+	0.612271i
$334$	−7.87574			−0.430941
$335$		−	3.70440i		−	0.202393i
$336$	−6.18523	−	2.27131i	−0.337432	−	0.123910i
$337$		−	21.0505i		−	1.14669i	−0.819313	−	0.573346i	$-0.805645\pi$
$337$		−	21.0505i		−	1.14669i	0.819313	−	0.573346i	$-0.194355\pi$
$338$		−	2.76537i		−	0.150416i
$339$	5.57771	−	15.1892i	0.302940	−	0.824965i
$340$	−4.76053			−0.258176
$341$	12.9652			0.702106
$342$	−4.30647	−	3.65577i	−0.232867	−	0.197681i
$343$		−	1.79555i		−	0.0969506i
$344$	6.96125			0.375326
$345$	8.16116	+	1.54773i	0.439382	+	0.0833271i
$346$	2.12595			0.114292
$347$			6.08253i			0.326528i	0.986582	+	0.163264i	$0.0522022\pi$
$347$			6.08253i			0.326528i	−0.986582	+	0.163264i	$0.947798\pi$
$348$	−12.2037	−	4.48137i	−0.654185	−	0.240227i
$349$	1.98670			0.106345			0.0531727	−	0.998585i	$-0.483067\pi$
$349$	1.98670			0.106345			0.0531727	+	0.998585i	$0.483067\pi$
$350$	−3.80421			−0.203343
$351$	14.4671	−	8.18774i	0.772195	−	0.437029i
$352$		−	3.48413i		−	0.185705i
$353$			8.53950i			0.454512i	0.973835	+	0.227256i	$0.0729754\pi$
$353$			8.53950i			0.454512i	−0.973835	+	0.227256i	$0.927025\pi$
$354$	4.39219	−	11.9608i	0.233442	−	0.635711i
$355$		−	9.92565i		−	0.526799i
$356$	11.0510			0.585700
$357$	29.4450	+	10.8126i	1.55839	+	0.572266i
$358$	12.4614			0.658606
$359$	−27.5129			−1.45207			−0.726037	−	0.687656i	$-0.758640\pi$
$359$	−27.5129			−1.45207			−0.726037	+	0.687656i	$0.758640\pi$
$360$	1.94149	−	2.28706i	0.102325	−	0.120538i
$361$	15.4544			0.813390
$362$	−10.2922			−0.540945
$363$	−0.680127	+	1.85212i	−0.0356974	+	0.0972112i
$364$		−	12.1703i		−	0.637895i
$365$	−3.18473			−0.166696
$366$	−1.98183	+	5.39692i	−0.103592	+	0.282101i
$367$			31.2664i			1.63209i	0.577986	+	0.816046i	$0.303839\pi$
$367$			31.2664i			1.63209i	−0.577986	+	0.816046i	$0.696161\pi$
$368$	0.785398	+	4.73108i	0.0409417	+	0.246625i
$369$	7.74774	−	9.12678i	0.403331	−	0.475121i
$370$	−4.88527			−0.253973
$371$		−	26.9108i		−	1.39714i
$372$	−2.22176	+	6.05031i	−0.115193	+	0.313694i
$373$		−	13.5799i		−	0.703138i	−0.936162	−	0.351569i	$-0.885648\pi$
$373$		−	13.5799i		−	0.703138i	0.936162	−	0.351569i	$-0.114352\pi$
$374$			16.5863i			0.857657i
$375$	0.597052	−	1.62589i	0.0308317	−	0.0839608i
$376$	−10.8399			−0.559025
$377$		−	24.0123i		−	1.23670i
$378$	−17.2032	+	9.73626i	−0.884835	+	0.500779i
$379$		−	31.3374i		−	1.60970i	−0.593482	−	0.804848i	$-0.702247\pi$
$379$		−	31.3374i		−	1.60970i	0.593482	−	0.804848i	$-0.297753\pi$
$380$		−	1.88297i		−	0.0965945i
$381$	9.36454	−	25.5015i	0.479760	−	1.30648i
$382$			22.5181i			1.15213i
$383$	−11.1675			−0.570632			−0.285316	−	0.958433i	$-0.592099\pi$
$383$	−11.1675			−0.570632			−0.285316	+	0.958433i	$0.592099\pi$
$384$	1.62589	+	0.597052i	0.0829710	+	0.0304682i
$385$			13.2543i			0.675504i
$386$			12.3106i			0.626591i
$387$	13.5152	−	15.9208i	0.687015	−	0.809299i
$388$			2.98155i			0.151365i
$389$	−12.5852			−0.638093			−0.319046	−	0.947739i	$-0.603363\pi$
$389$	−12.5852			−0.638093			−0.319046	+	0.947739i	$0.603363\pi$
$390$	5.20149	+	1.91007i	0.263388	+	0.0967200i
$391$	−3.73891	−	22.5225i	−0.189085	−	1.13901i
$392$			7.47199i			0.377393i
$393$	−17.1537	−	6.29910i	−0.865291	−	0.317748i
$394$	−0.321151			−0.0161793
$395$			10.1264i			0.509513i
$396$	−7.96840	−	6.76439i	−0.400427	−	0.339923i
$397$	12.3539			0.620024			0.310012	−	0.950733i	$-0.399667\pi$
$397$	12.3539			0.620024			0.310012	+	0.950733i	$0.399667\pi$
$398$	14.6664			0.735162
$399$	−4.27682	+	11.6466i	−0.214109	+	0.583061i
$400$	1.00000			0.0500000
$401$	27.2453			1.36057			0.680283	−	0.732950i	$-0.261857\pi$
$401$	27.2453			1.36057			0.680283	+	0.732950i	$0.261857\pi$
$402$	2.21172	−	6.02295i	0.110311	−	0.300398i
$403$	−11.9048			−0.593020
$404$		−	9.15523i		−	0.455490i
$405$	−1.46126	−	8.88058i	−0.0726105	−	0.441280i
$406$			28.5537i			1.41710i
$407$			17.0209i			0.843695i
$408$	−7.74011	−	2.84229i	−0.383193	−	0.140714i
$409$	−27.3294			−1.35135			−0.675675	−	0.737200i	$-0.736148\pi$
$409$	−27.3294			−1.35135			−0.675675	+	0.737200i	$0.736148\pi$
$410$	3.99062			0.197083
$411$	9.70890	−	26.4393i	0.478905	−	1.30415i
$412$		−	16.0594i		−	0.791190i
$413$	−27.9855			−1.37708
$414$	12.3451	+	7.38909i	0.606728	+	0.363154i
$415$	12.8088			0.628759
$416$			3.19916i			0.156852i
$417$	13.9373	−	37.9540i	0.682511	−	1.85862i
$418$	−6.56052			−0.320885
$419$	15.1668			0.740949			0.370474	−	0.928843i	$-0.379195\pi$
$419$	15.1668			0.740949			0.370474	+	0.928843i	$0.379195\pi$
$420$	−6.18523	−	2.27131i	−0.301809	−	0.110829i
$421$		−	34.7728i		−	1.69472i	−0.531018	−	0.847361i	$-0.678190\pi$
$421$		−	34.7728i		−	1.69472i	0.531018	−	0.847361i	$-0.321810\pi$
$422$			12.3848i			0.602882i
$423$	−21.0455	+	24.7915i	−1.02327	+	1.20540i
$424$			7.07396i			0.343542i
$425$	−4.76053			−0.230920
$426$	5.92613	−	16.1380i	0.287122	−	0.781891i
$427$	12.6275			0.611088
$428$	−17.6195			−0.851669
$429$	6.65491	−	18.1227i	0.321302	−	0.874970i
$430$	6.96125			0.335701
$431$	−5.86936			−0.282717			−0.141358	−	0.989958i	$-0.545147\pi$
$431$	−5.86936			−0.282717			−0.141358	+	0.989958i	$0.545147\pi$
$432$	4.52214	−	2.55934i	0.217572	−	0.123136i
$433$			30.4782i			1.46469i	0.680935	+	0.732344i	$0.261574\pi$
$433$			30.4782i			1.46469i	−0.680935	+	0.732344i	$0.738426\pi$
$434$	14.1563			0.679524
$435$	−12.2037	−	4.48137i	−0.585121	−	0.214865i
$436$		−	12.7316i		−	0.609733i
$437$	8.90850	−	1.47888i	0.426152	−	0.0707446i
$438$	−5.17803	−	1.90145i	−0.247416	−	0.0908548i
$439$	20.8965			0.997334			0.498667	−	0.866794i	$-0.333823\pi$
$439$	20.8965			0.997334			0.498667	+	0.866794i	$0.333823\pi$
$440$		−	3.48413i		−	0.166099i
$441$	17.0889	+	14.5068i	0.813756	+	0.690799i
$442$		−	15.2297i		−	0.724403i
$443$			38.9828i			1.85213i	0.377365	+	0.926065i	$0.376830\pi$
$443$			38.9828i			1.85213i	−0.377365	+	0.926065i	$0.623170\pi$
$444$	−7.94293	−	2.91676i	−0.376955	−	0.138423i
$445$	11.0510			0.523866
$446$			5.19193i			0.245845i
$447$	−4.18714	+	11.4024i	−0.198045	+	0.539317i
$448$		−	3.80421i		−	0.179732i
$449$		−	8.98085i		−	0.423832i	−0.977288	−	0.211916i	$-0.932030\pi$
$449$		−	8.98085i		−	0.423832i	0.977288	−	0.211916i	$-0.0679704\pi$
$450$	1.94149	−	2.28706i	0.0915226	−	0.107813i
$451$		−	13.9038i		−	0.654706i
$452$	9.34208			0.439414
$453$	8.54699	−	23.2752i	0.401572	−	1.09356i
$454$			16.7298i			0.785170i
$455$		−	12.1703i		−	0.570551i
$456$	1.12423	−	3.06151i	0.0526471	−	0.143369i
$457$		−	5.68472i		−	0.265920i	−0.991121	−	0.132960i	$-0.957552\pi$
$457$		−	5.68472i		−	0.265920i	0.991121	−	0.132960i	$-0.0424482\pi$
$458$	−18.2985			−0.855031
$459$	−21.5278	+	12.1838i	−1.00483	+	0.568692i
$460$	0.785398	+	4.73108i	0.0366194	+	0.220588i
$461$			23.4737i			1.09328i	0.837368	+	0.546640i	$0.184093\pi$
$461$			23.4737i			1.09328i	−0.837368	+	0.546640i	$0.815907\pi$
$462$	−7.91354	+	21.5501i	−0.368171	+	1.00260i
$463$	−41.4142			−1.92468			−0.962339	−	0.271851i	$-0.912364\pi$
$463$	−41.4142			−1.92468			−0.962339	+	0.271851i	$0.912364\pi$
$464$		−	7.50582i		−	0.348449i
$465$	−2.22176	+	6.05031i	−0.103032	+	0.280576i
$466$	−6.92267			−0.320686
$467$	28.5266			1.32006			0.660028	−	0.751241i	$-0.270545\pi$
$467$	28.5266			1.32006			0.660028	+	0.751241i	$0.270545\pi$
$468$	7.31666	+	6.21113i	0.338213	+	0.287109i
$469$	−14.0923			−0.650722
$470$	−10.8399			−0.500007
$471$	−26.6458	−	9.78475i	−1.22778	−	0.450858i
$472$	7.35646			0.338609
$473$		−	24.2539i		−	1.11519i
$474$	−6.04597	+	16.4644i	−0.277701	+	0.756235i
$475$		−	1.88297i		−	0.0863967i
$476$			18.1100i			0.830072i
$477$	16.1785	+	13.7340i	0.740765	+	0.628836i
$478$	−2.10045			−0.0960724
$479$	3.94290			0.180155			0.0900777	−	0.995935i	$-0.471288\pi$
$479$	3.94290			0.180155			0.0900777	+	0.995935i	$0.471288\pi$
$480$	1.62589	+	0.597052i	0.0742115	+	0.0272516i
$481$		−	15.6288i		−	0.712610i
$482$	0.0258170			0.00117593
$483$	5.88789	−	31.0467i	0.267908	−	1.41268i
$484$	−1.13914			−0.0517792
$485$			2.98155i			0.135385i
$486$	2.92632	−	15.3113i	0.132741	−	0.694536i
$487$	1.02454			0.0464262			0.0232131	−	0.999731i	$-0.492610\pi$
$487$	1.02454			0.0464262			0.0232131	+	0.999731i	$0.492610\pi$
$488$	−3.31936			−0.150260
$489$	−3.14363	+	8.56074i	−0.142160	+	0.387130i
$490$			7.47199i			0.337550i
$491$		−	12.3413i		−	0.556954i	−0.960443	−	0.278477i	$-0.910170\pi$
$491$		−	12.3413i		−	0.556954i	0.960443	−	0.278477i	$-0.0898296\pi$
$492$	6.48833	+	2.38261i	0.292516	+	0.107416i
$493$			35.7317i			1.60927i
$494$	6.02393			0.271030
$495$	−7.96840	−	6.76439i	−0.358153	−	0.304037i
$496$	−3.72122			−0.167088
$497$	−37.7592			−1.69373
$498$	20.8257	+	7.64752i	0.933224	+	0.342694i
$499$	−29.3987			−1.31607			−0.658033	−	0.752989i	$-0.728611\pi$
$499$	−29.3987			−1.31607			−0.658033	+	0.752989i	$0.728611\pi$
$500$	1.00000			0.0447214
$501$	−12.8051	−	4.70223i	−0.572090	−	0.210080i
$502$			7.11730i			0.317661i
$503$	−27.8709			−1.24270			−0.621350	−	0.783533i	$-0.713416\pi$
$503$	−27.8709			−1.24270			−0.621350	+	0.783533i	$0.713416\pi$
$504$	−8.70044	−	7.38582i	−0.387548	−	0.328990i
$505$		−	9.15523i		−	0.407402i
$506$	16.4837	−	2.73642i	0.732790	−	0.121649i
$507$	1.65107	−	4.49620i	0.0733267	−	0.199683i
$508$	15.6846			0.695892
$509$			33.5429i			1.48676i	0.668868	+	0.743381i	$0.266779\pi$
$509$			33.5429i			1.48676i	−0.668868	+	0.743381i	$0.733221\pi$
$510$	−7.74011	−	2.84229i	−0.342738	−	0.125859i
$511$			12.1154i			0.535952i
$512$			1.00000i			0.0441942i
$513$	−4.81917	−	8.51507i	−0.212772	−	0.375950i
$514$	−5.27667			−0.232744
$515$		−	16.0594i		−	0.707662i
$516$	11.3183	+	4.15623i	0.498258	+	0.182968i
$517$			37.7676i			1.66102i
$518$			18.5846i			0.816559i
$519$	3.45657	+	1.26930i	0.151727	+	0.0557163i
$520$			3.19916i			0.140292i
$521$	4.78488			0.209629			0.104815	−	0.994492i	$-0.466575\pi$
$521$	4.78488			0.209629			0.104815	+	0.994492i	$0.466575\pi$
$522$	−17.1662	−	14.5725i	−0.751346	−	0.637819i
$523$		−	16.7160i		−	0.730939i	−0.930823	−	0.365470i	$-0.880908\pi$
$523$		−	16.7160i		−	0.730939i	0.930823	−	0.365470i	$-0.119092\pi$
$524$		−	10.5503i		−	0.460894i
$525$	−6.18523	−	2.27131i	−0.269946	−	0.0991281i
$526$			13.6544i			0.595361i
$527$	17.7150			0.771677
$528$	2.08021	−	5.66482i	0.0905294	−	0.246530i
$529$	−21.7663	+	7.43156i	−0.946361	+	0.323111i
$530$			7.07396i			0.307273i
$531$	14.2825	−	16.8247i	0.619807	−	0.730128i
$532$	−7.16322			−0.310565
$533$			12.7666i			0.552985i
$534$	17.9677	+	6.59801i	0.777538	+	0.285524i
$535$	−17.6195			−0.761756
$536$	3.70440			0.160006
$537$	20.2609	+	7.44011i	0.874323	+	0.321064i
$538$	14.9128			0.642936
$539$	26.0334			1.12134
$540$	4.52214	−	2.55934i	0.194602	−	0.110136i
$541$	−16.8464			−0.724284			−0.362142	−	0.932123i	$-0.617954\pi$
$541$	−16.8464			−0.724284			−0.362142	+	0.932123i	$0.617954\pi$
$542$		−	22.4734i		−	0.965314i
$543$	−16.7340	−	6.14497i	−0.718124	−	0.263706i
$544$		−	4.76053i		−	0.204106i
$545$		−	12.7316i		−	0.545361i
$546$	7.26629	−	19.7876i	0.310968	−	0.846829i
$547$	17.5645			0.751005			0.375503	−	0.926821i	$-0.377470\pi$
$547$	17.5645			0.751005			0.375503	+	0.926821i	$0.377470\pi$
$548$	16.2614			0.694652
$549$	−6.44449	+	7.59156i	−0.275044	+	0.324000i
$550$		−	3.48413i		−	0.148564i
$551$	−14.1333			−0.602097
$552$	−1.54773	+	8.16116i	−0.0658759	+	0.347362i
$553$	38.5228			1.63816
$554$		−	25.7920i		−	1.09580i
$555$	−7.94293	−	2.91676i	−0.337158	−	0.123810i
$556$	23.3435			0.989984
$557$	6.58788			0.279137			0.139569	−	0.990212i	$-0.455428\pi$
$557$	6.58788			0.279137			0.139569	+	0.990212i	$0.455428\pi$
$558$	−7.22470	+	8.51065i	−0.305846	+	0.360284i
$559$			22.2702i			0.941928i
$560$		−	3.80421i		−	0.160757i
$561$	−9.90288	+	26.9675i	−0.418100	+	1.13857i
$562$			5.19176i			0.219001i
$563$	32.6960			1.37797			0.688987	−	0.724774i	$-0.258056\pi$
$563$	32.6960			1.37797			0.688987	+	0.724774i	$0.258056\pi$
$564$	−17.6245	−	6.47199i	−0.742126	−	0.272520i
$565$	9.34208			0.393024
$566$	6.60335			0.277560
$567$	−33.7836	+	5.55893i	−1.41878	+	0.233453i
$568$	9.92565			0.416471
$569$	0.358940			0.0150476			0.00752378	−	0.999972i	$-0.497605\pi$
$569$	0.358940			0.0150476			0.00752378	+	0.999972i	$0.497605\pi$
$570$	1.12423	−	3.06151i	0.0470890	−	0.128233i
$571$			32.0526i			1.34136i	0.741747	+	0.670680i	$0.233998\pi$
$571$			32.0526i			1.34136i	−0.741747	+	0.670680i	$0.766002\pi$
$572$	11.1463			0.466049
$573$	−13.4445	+	36.6120i	−0.561651	+	1.52949i
$574$		−	15.1812i		−	0.633649i
$575$	0.785398	+	4.73108i	0.0327533	+	0.197300i
$576$	2.28706	+	1.94149i	0.0952940	+	0.0808953i
$577$	4.12555			0.171749			0.0858745	−	0.996306i	$-0.472632\pi$
$577$	4.12555			0.171749			0.0858745	+	0.996306i	$0.472632\pi$
$578$			5.66264i			0.235535i
$579$	−7.35005	+	20.0157i	−0.305458	+	0.831822i
$580$		−	7.50582i		−	0.311662i
$581$		−	48.7273i		−	2.02155i
$582$	−1.78014	+	4.84768i	−0.0737892	+	0.200943i
$583$	24.6466			1.02076
$584$		−	3.18473i		−	0.131785i
$585$	7.31666	+	6.21113i	0.302507	+	0.256799i
$586$			30.6756i			1.26720i
$587$			41.6293i			1.71822i	0.511787	+	0.859112i	$0.328984\pi$
$587$			41.6293i			1.71822i	−0.511787	+	0.859112i	$0.671016\pi$
$588$	−4.46117	+	12.1487i	−0.183976	+	0.501002i
$589$			7.00696i			0.288717i
$590$	7.35646			0.302861
$591$	−0.522157	−	0.191744i	−0.0214787	−	0.00788729i
$592$		−	4.88527i		−	0.200783i
$593$		−	7.96907i		−	0.327250i	−0.986523	−	0.163625i	$-0.947681\pi$
$593$		−	7.96907i		−	0.327250i	0.986523	−	0.163625i	$-0.0523188\pi$
$594$	−8.91707	−	15.7557i	−0.365872	−	0.646465i
$595$			18.1100i			0.742439i
$596$	−7.01303			−0.287265
$597$	23.8461	+	8.75663i	0.975955	+	0.358385i
$598$	−15.1355	+	2.51261i	−0.618936	+	0.102748i
$599$			18.0460i			0.737338i	0.929561	+	0.368669i	$0.120186\pi$
$599$			18.0460i			0.737338i	−0.929561	+	0.368669i	$0.879814\pi$
$600$	1.62589	+	0.597052i	0.0663768	+	0.0243746i
$601$	−14.6307			−0.596800			−0.298400	−	0.954441i	$-0.596453\pi$
$601$	−14.6307			−0.596800			−0.298400	+	0.954441i	$0.596453\pi$
$602$		−	26.4820i		−	1.07933i
$603$	7.19204	−	8.47217i	0.292882	−	0.345013i
$604$	14.3153			0.582481
$605$	−1.13914			−0.0463127
$606$	5.46615	−	14.8854i	0.222047	−	0.604679i
$607$	−6.64798			−0.269833			−0.134917	−	0.990857i	$-0.543077\pi$
$607$	−6.64798			−0.269833			−0.134917	+	0.990857i	$0.543077\pi$
$608$	1.88297			0.0763647
$609$	−17.0481	+	46.4253i	−0.690822	+	1.88125i
$610$	−3.31936			−0.134397
$611$		−	34.6786i		−	1.40295i
$612$	−10.8876	−	9.24251i	−0.440105	−	0.373606i
$613$			3.79751i			0.153380i	0.997055	+	0.0766899i	$0.0244351\pi$
$613$			3.79751i			0.153380i	−0.997055	+	0.0766899i	$0.975565\pi$
$614$			19.3975i			0.782819i
$615$	6.48833	+	2.38261i	0.261635	+	0.0960761i
$616$	−13.2543			−0.534033
$617$	10.0968			0.406483			0.203242	−	0.979129i	$-0.434852\pi$
$617$	10.0968			0.406483			0.203242	+	0.979129i	$0.434852\pi$
$618$	9.58831	−	26.1109i	0.385698	−	1.05033i
$619$			19.8519i			0.797917i	0.916969	+	0.398959i	$0.130628\pi$
$619$			19.8519i			0.797917i	−0.916969	+	0.398959i	$0.869372\pi$
$620$	−3.72122			−0.149448
$621$	15.6601	+	19.3845i	0.628420	+	0.777875i
$622$	16.8790			0.676785
$623$		−	42.0402i		−	1.68430i
$624$	−1.91007	+	5.20149i	−0.0764638	+	0.208226i
$625$	1.00000			0.0400000
$626$	26.7256			1.06817
$627$	−10.6667	−	3.91697i	−0.425987	−	0.156429i
$628$		−	16.3884i		−	0.653970i
$629$			23.2565i			0.927296i
$630$	−8.70044	−	7.38582i	−0.346634	−	0.294258i
$631$			32.8990i			1.30969i	0.755764	+	0.654845i	$0.227266\pi$
$631$			32.8990i			1.30969i	−0.755764	+	0.654845i	$0.772734\pi$
$632$	−10.1264			−0.402805
$633$	−7.39437	+	20.1363i	−0.293900	+	0.800348i
$634$	−7.55156			−0.299911
$635$	15.6846			0.622425
$636$	−4.22352	+	11.5015i	−0.167474	+	0.456064i
$637$	−23.9041			−0.947115
$638$	−26.1512			−1.03534
$639$	19.2705	−	22.7005i	0.762330	−	0.898019i
$640$			1.00000i			0.0395285i
$641$	−3.24280			−0.128083			−0.0640415	−	0.997947i	$-0.520399\pi$
$641$	−3.24280			−0.128083			−0.0640415	+	0.997947i	$0.520399\pi$
$642$	−28.6474	−	10.5197i	−1.13062	−	0.415181i
$643$			29.2876i			1.15499i	0.816395	+	0.577494i	$0.195969\pi$
$643$			29.2876i			1.15499i	−0.816395	+	0.577494i	$0.804031\pi$
$644$	17.9980	−	2.98781i	0.709221	−	0.117736i
$645$	11.3183	+	4.15623i	0.445656	+	0.163651i
$646$	−8.96395			−0.352682
$647$		−	1.01376i		−	0.0398551i	−0.999801	−	0.0199276i	$-0.993656\pi$
$647$		−	1.01376i		−	0.0398551i	0.999801	−	0.0199276i	$-0.00634356\pi$
$648$	8.88058	−	1.46126i	0.348862	−	0.0574037i
$649$		−	25.6309i		−	1.00610i
$650$			3.19916i			0.125481i
$651$	23.0166	+	8.45205i	0.902093	+	0.331262i
$652$	−5.26525			−0.206203
$653$			1.94489i			0.0761093i	0.999276	+	0.0380546i	$0.0121161\pi$
$653$			1.94489i			0.0761093i	−0.999276	+	0.0380546i	$0.987884\pi$
$654$	7.60143	−	20.7002i	0.297239	−	0.809442i
$655$		−	10.5503i		−	0.412236i
$656$			3.99062i			0.155808i
$657$	−7.28365	−	6.18311i	−0.284162	−	0.241226i
$658$			41.2372i			1.60759i
$659$	−40.4721			−1.57657			−0.788285	−	0.615310i	$-0.789031\pi$
$659$	−40.4721			−1.57657			−0.788285	+	0.615310i	$0.789031\pi$
$660$	2.08021	−	5.66482i	0.0809719	−	0.220503i
$661$			18.3138i			0.712324i	0.934424	+	0.356162i	$0.115915\pi$
$661$			18.3138i			0.712324i	−0.934424	+	0.356162i	$0.884085\pi$
$662$			25.3475i			0.985159i
$663$	9.09293	−	24.7619i	0.353140	−	0.961671i
$664$			12.8088i			0.497078i
$665$	−7.16322			−0.277778
$666$	−11.1729	−	9.48469i	−0.432941	−	0.367524i
$667$	35.5107	−	5.89505i	1.37498	−	0.228257i
$668$		−	7.87574i		−	0.304722i
$669$	−3.09985	+	8.44152i	−0.119847	+	0.326368i
$670$	3.70440			0.143113
$671$			11.5651i			0.446464i
$672$	2.27131	−	6.18523i	0.0876177	−	0.238601i
$673$	31.8250			1.22676			0.613381	−	0.789787i	$-0.289809\pi$
$673$	31.8250			1.22676			0.613381	+	0.789787i	$0.289809\pi$
$674$	21.0505			0.810834
$675$	4.52214	−	2.55934i	0.174057	−	0.0985091i
$676$	2.76537			0.106360
$677$	18.5947			0.714654			0.357327	−	0.933979i	$-0.383688\pi$
$677$	18.5947			0.714654			0.357327	+	0.933979i	$0.383688\pi$
$678$	15.1892	+	5.57771i	0.583339	+	0.214211i
$679$	11.3424			0.435282
$680$		−	4.76053i		−	0.182558i
$681$	−9.98859	+	27.2009i	−0.382763	+	1.04234i
$682$			12.9652i			0.496464i
$683$			12.1437i			0.464667i	0.972636	+	0.232334i	$0.0746361\pi$
$683$			12.1437i			0.464667i	−0.972636	+	0.232334i	$0.925364\pi$
$684$	3.65577	−	4.30647i	0.139782	−	0.164662i
$685$	16.2614			0.621316
$686$	1.79555			0.0685544
$687$	−29.7514	−	10.9251i	−1.13509	−	0.416820i
$688$			6.96125i			0.265395i
$689$	−22.6307			−0.862162
$690$	−1.54773	+	8.16116i	−0.0589212	+	0.310690i
$691$	−33.6060			−1.27843			−0.639217	−	0.769027i	$-0.720741\pi$
$691$	−33.6060			−1.27843			−0.639217	+	0.769027i	$0.720741\pi$
$692$			2.12595i			0.0808165i
$693$	−25.7331	+	30.3134i	−0.977521	+	1.15151i
$694$	−6.08253			−0.230890
$695$	23.3435			0.885469
$696$	4.48137	−	12.2037i	0.169866	−	0.462579i
$697$		−	18.9975i		−	0.719581i
$698$			1.98670i			0.0751975i
$699$	−11.2555	−	4.13319i	−0.425723	−	0.156332i
$700$		−	3.80421i		−	0.143786i
$701$	−36.0032			−1.35982			−0.679911	−	0.733295i	$-0.737982\pi$
$701$	−36.0032			−1.35982			−0.679911	+	0.733295i	$0.737982\pi$
$702$	8.18774	+	14.4671i	0.309026	+	0.546024i
$703$	−9.19883			−0.346941
$704$	3.48413			0.131313
$705$	−17.6245	−	6.47199i	−0.663778	−	0.243749i
$706$	−8.53950			−0.321388
$707$	−34.8284			−1.30986
$708$	11.9608	+	4.39219i	0.449515	+	0.165069i
$709$		−	37.0502i		−	1.39145i	−0.718308	−	0.695725i	$-0.755083\pi$
$709$		−	37.0502i		−	1.39145i	0.718308	−	0.695725i	$-0.244917\pi$
$710$	9.92565			0.372503
$711$	−19.6602	+	23.1596i	−0.737316	+	0.868552i
$712$			11.0510i			0.414153i
$713$	−2.92264	−	17.6054i	−0.109454	−	0.659328i
$714$	−10.8126	+	29.4450i	−0.404653	+	1.10195i
$715$	11.1463			0.416847
$716$			12.4614i			0.465704i
$717$	−3.41511	−	1.25408i	−0.127540	−	0.0468345i
$718$		−	27.5129i		−	1.02677i
$719$			22.1798i			0.827167i	0.910466	+	0.413583i	$0.135723\pi$
$719$			22.1798i			0.827167i	−0.910466	+	0.413583i	$0.864277\pi$
$720$	2.28706	+	1.94149i	0.0852336	+	0.0723549i
$721$	−61.0933			−2.27523
$722$			15.4544i			0.575154i
$723$	0.0419757	+	0.0154141i	0.00156109	+	0.000573257i
$724$		−	10.2922i		−	0.382506i
$725$		−	7.50582i		−	0.278759i
$726$	−1.85212	−	0.680127i	−0.0687387	−	0.0252419i
$727$			25.7337i			0.954410i	0.878792	+	0.477205i	$0.158350\pi$
$727$			25.7337i			0.954410i	−0.878792	+	0.477205i	$0.841650\pi$
$728$	12.1703			0.451060
$729$	13.8995	−	23.1474i	0.514798	−	0.857311i
$730$		−	3.18473i		−	0.117872i
$731$		−	33.1392i		−	1.22570i
$732$	−5.39692	−	1.98183i	−0.199476	−	0.0732505i
$733$			29.9581i			1.10653i	0.833006	+	0.553264i	$0.186618\pi$
$733$			29.9581i			1.10653i	−0.833006	+	0.553264i	$0.813382\pi$
$734$	−31.2664			−1.15406
$735$	−4.46117	+	12.1487i	−0.164553	+	0.448110i
$736$	−4.73108	+	0.785398i	−0.174390	+	0.0289501i
$737$		−	12.9066i		−	0.475420i
$738$	9.12678	+	7.74774i	0.335961	+	0.285198i
$739$	−5.95900			−0.219206			−0.109603	−	0.993975i	$-0.534958\pi$
$739$	−5.95900			−0.219206			−0.109603	+	0.993975i	$0.534958\pi$
$740$		−	4.88527i		−	0.179586i
$741$	9.79427	+	3.59660i	0.359802	+	0.132125i
$742$	26.9108			0.987926
$743$	20.9476			0.768494			0.384247	−	0.923230i	$-0.374461\pi$
$743$	20.9476			0.768494			0.384247	+	0.923230i	$0.374461\pi$
$744$	−6.05031	−	2.22176i	−0.221815	−	0.0814539i
$745$	−7.01303			−0.256937
$746$	13.5799			0.497194
$747$	29.2944	+	24.8681i	1.07183	+	0.909877i
$748$	−16.5863			−0.606455
$749$			67.0281i			2.44915i
$750$	1.62589	+	0.597052i	0.0593692	+	0.0218013i
$751$		−	30.8923i		−	1.12727i	−0.826022	−	0.563637i	$-0.809402\pi$
$751$		−	30.8923i		−	1.12727i	0.826022	−	0.563637i	$-0.190598\pi$
$752$		−	10.8399i		−	0.395291i
$753$	−4.24940	+	11.5720i	−0.154857	+	0.421706i
$754$	24.0123			0.874477
$755$	14.3153			0.520987
$756$	−9.73626	−	17.2032i	−0.354104	−	0.625673i
$757$		−	15.9263i		−	0.578851i	−0.957200	−	0.289426i	$-0.906536\pi$
$757$		−	15.9263i		−	0.578851i	0.957200	−	0.289426i	$-0.0934643\pi$
$758$	31.3374			1.13823
$759$	28.4345	+	5.39250i	1.03211	+	0.195735i
$760$	1.88297			0.0683026
$761$		−	23.2817i		−	0.843960i	−0.906605	−	0.421980i	$-0.861335\pi$
$761$		−	23.2817i		−	0.843960i	0.906605	−	0.421980i	$-0.138665\pi$
$762$	25.5015	+	9.36454i	0.923822	+	0.339241i
$763$	−48.4336			−1.75341
$764$	−22.5181			−0.814676
$765$	−10.8876	−	9.24251i	−0.393642	−	0.334163i
$766$		−	11.1675i		−	0.403498i
$767$			23.5345i			0.849782i
$768$	−0.597052	+	1.62589i	−0.0215443	+	0.0586694i
$769$		−	33.0800i		−	1.19289i	−0.802653	−	0.596447i	$-0.796579\pi$
$769$		−	33.0800i		−	1.19289i	0.802653	−	0.596447i	$-0.203421\pi$
$770$	−13.2543			−0.477653
$771$	−8.57931	−	3.15045i	−0.308976	−	0.113461i
$772$	−12.3106			−0.443067
$773$	37.7407			1.35744			0.678720	−	0.734397i	$-0.262535\pi$
$773$	37.7407			1.35744			0.678720	+	0.734397i	$0.262535\pi$
$774$	15.9208	+	13.5152i	0.572261	+	0.485793i
$775$	−3.72122			−0.133670
$776$	−2.98155			−0.107031
$777$	−11.0960	+	30.2165i	−0.398066	+	1.08401i
$778$		−	12.5852i		−	0.451200i
$779$	7.51424			0.269225
$780$	−1.91007	+	5.20149i	−0.0683913	+	0.186243i
$781$		−	34.5822i		−	1.23745i
$782$	22.5225	−	3.73891i	0.805402	−	0.133703i
$783$	−19.2100	−	33.9424i	−0.686508	−	1.21300i
$784$	−7.47199			−0.266857
$785$		−	16.3884i		−	0.584928i
$786$	6.29910	−	17.1537i	0.224682	−	0.611853i
$787$			10.3941i			0.370508i	0.982691	+	0.185254i	$0.0593108\pi$
$787$			10.3941i			0.370508i	−0.982691	+	0.185254i	$0.940689\pi$
$788$		−	0.321151i		−	0.0114405i
$789$	−8.15240	+	22.2006i	−0.290233	+	0.790363i
$790$	−10.1264			−0.360280
$791$		−	35.5392i		−	1.26363i
$792$	6.76439	−	7.96840i	0.240362	−	0.283145i
$793$		−	10.6192i		−	0.377097i
$794$			12.3539i			0.438423i
$795$	−4.22352	+	11.5015i	−0.149793	+	0.407916i
$796$			14.6664i			0.519838i
$797$	18.4049			0.651934			0.325967	−	0.945381i	$-0.394310\pi$
$797$	18.4049			0.651934			0.325967	+	0.945381i	$0.394310\pi$
$798$	−11.6466	−	4.27682i	−0.412286	−	0.151398i
$799$			51.6037i			1.82561i
$800$			1.00000i			0.0353553i
$801$	25.2742	+	21.4553i	0.893020	+	0.758086i
$802$			27.2453i			0.962065i
$803$	−11.0960			−0.391569
$804$	6.02295	+	2.21172i	0.212413	+	0.0780013i
$805$	17.9980	−	2.98781i	0.634347	−	0.105307i
$806$		−	11.9048i		−	0.419328i
$807$	24.2466	+	8.90372i	0.853521	+	0.313426i
$808$	9.15523			0.322080
$809$		−	38.2278i		−	1.34402i	−0.740543	−	0.672009i	$-0.765431\pi$
$809$		−	38.2278i		−	1.34402i	0.740543	−	0.672009i	$-0.234569\pi$
$810$	8.88058	−	1.46126i	0.312032	−	0.0513434i
$811$	−34.0643			−1.19616			−0.598080	−	0.801436i	$-0.704070\pi$
$811$	−34.0643			−1.19616			−0.598080	+	0.801436i	$0.704070\pi$
$812$	−28.5537			−1.00204
$813$	13.4178	−	36.5393i	0.470582	−	1.28149i
$814$	−17.0209			−0.596582
$815$	−5.26525			−0.184434
$816$	2.84229	−	7.74011i	0.0994999	−	0.270958i
$817$	13.1079			0.458586
$818$		−	27.3294i		−	0.955549i
$819$	23.6284	−	27.8341i	0.825644	−	0.972602i
$820$			3.99062i			0.139359i
$821$			10.8083i			0.377212i	0.982053	+	0.188606i	$0.0603969\pi$
$821$			10.8083i			0.377212i	−0.982053	+	0.188606i	$0.939603\pi$
$822$	26.4393	+	9.70890i	0.922176	+	0.338637i
$823$	52.1001			1.81610			0.908048	−	0.418865i	$-0.137572\pi$
$823$	52.1001			1.81610			0.908048	+	0.418865i	$0.137572\pi$
$824$	16.0594			0.559456
$825$	2.08021	−	5.66482i	0.0724235	−	0.197224i
$826$		−	27.9855i		−	0.973740i
$827$	−56.5967			−1.96806			−0.984030	−	0.178005i	$-0.943036\pi$
$827$	−56.5967			−1.96806			−0.984030	+	0.178005i	$0.943036\pi$
$828$	−7.38909	+	12.3451i	−0.256789	+	0.429022i
$829$	32.2985			1.12177			0.560887	−	0.827892i	$-0.310460\pi$
$829$	32.2985			1.12177			0.560887	+	0.827892i	$0.310460\pi$
$830$			12.8088i			0.444600i
$831$	15.3992	−	41.9351i	0.534192	−	1.45471i
$832$	−3.19916			−0.110911
$833$	35.5706			1.23245
$834$	37.9540	+	13.9373i	1.31424	+	0.482608i
$835$		−	7.87574i		−	0.272551i
$836$		−	6.56052i		−	0.226900i
$837$	−16.8279	+	9.52388i	−0.581657	+	0.329193i
$838$			15.1668i			0.523930i
$839$	−28.9576			−0.999728			−0.499864	−	0.866104i	$-0.666617\pi$
$839$	−28.9576			−0.999728			−0.499864	+	0.866104i	$0.666617\pi$
$840$	2.27131	−	6.18523i	0.0783677	−	0.213411i
$841$	−27.3373			−0.942667
$842$	34.7728			1.19835
$843$	−3.09975	+	8.44124i	−0.106761	+	0.290732i
$844$	−12.3848			−0.426302
$845$	2.76537			0.0951317
$846$	−24.7915	−	21.0455i	−0.852349	−	0.723560i
$847$			4.33353i			0.148902i
$848$	−7.07396			−0.242921
$849$	10.7363	+	3.94255i	0.368470	+	0.135308i
$850$		−	4.76053i		−	0.163285i
$851$	23.1126	−	3.83688i	0.792290	−	0.131527i
$852$	16.1380	+	5.92613i	0.552880	+	0.203026i
$853$	7.15727			0.245060			0.122530	−	0.992465i	$-0.460899\pi$
$853$	7.15727			0.245060			0.122530	+	0.992465i	$0.460899\pi$
$854$			12.6275i			0.432105i
$855$	3.65577	−	4.30647i	0.125025	−	0.147278i
$856$		−	17.6195i		−	0.602221i
$857$			48.9333i			1.67153i	0.549089	+	0.835764i	$0.314975\pi$
$857$			48.9333i			1.67153i	−0.549089	+	0.835764i	$0.685025\pi$
$858$	18.1227	+	6.65491i	0.618697	+	0.227195i
$859$	25.7121			0.877286			0.438643	−	0.898661i	$-0.355459\pi$
$859$	25.7121			0.877286			0.438643	+	0.898661i	$0.355459\pi$
$860$			6.96125i			0.237377i
$861$	9.06394	−	24.6829i	0.308898	−	0.841192i
$862$		−	5.86936i		−	0.199911i
$863$			11.0725i			0.376912i	0.982082	+	0.188456i	$0.0603483\pi$
$863$			11.0725i			0.376912i	−0.982082	+	0.188456i	$0.939652\pi$
$864$	2.55934	+	4.52214i	0.0870705	+	0.153846i
$865$			2.12595i			0.0722845i
$866$	−30.4782			−1.03569
$867$	−3.38089	+	9.20685i	−0.114821	+	0.312681i
$868$			14.1563i			0.480496i
$869$			35.2816i			1.19685i
$870$	4.48137	−	12.2037i	0.151933	−	0.413743i
$871$			11.8510i			0.401554i
$872$	12.7316			0.431146
$873$	−5.78864	+	6.81897i	−0.195916	+	0.230787i
$874$	1.47888	+	8.90850i	0.0500240	+	0.301335i
$875$		−	3.80421i		−	0.128606i
$876$	1.90145	−	5.17803i	0.0642440	−	0.174949i
$877$	34.2809			1.15758			0.578791	−	0.815476i	$-0.303525\pi$
$877$	34.2809			1.15758			0.578791	+	0.815476i	$0.303525\pi$
$878$			20.8965i			0.705222i
$879$	−18.3150	+	49.8753i	−0.617748	+	1.68225i
$880$	3.48413			0.117450
$881$	21.4410			0.722367			0.361183	−	0.932495i	$-0.382373\pi$
$881$	21.4410			0.722367			0.361183	+	0.932495i	$0.382373\pi$
$882$	−14.5068	+	17.0889i	−0.488468	+	0.575412i
$883$	14.2372			0.479119			0.239560	−	0.970882i	$-0.422997\pi$
$883$	14.2372			0.479119			0.239560	+	0.970882i	$0.422997\pi$
$884$	15.2297			0.512230
$885$	11.9608	+	4.39219i	0.402059	+	0.147642i
$886$	−38.9828			−1.30965
$887$			54.3381i			1.82449i	0.409640	+	0.912247i	$0.365654\pi$
$887$			54.3381i			1.82449i	−0.409640	+	0.912247i	$0.634346\pi$
$888$	2.91676	−	7.94293i	0.0978801	−	0.266547i
$889$		−	59.6675i		−	2.00118i
$890$			11.0510i			0.370429i
$891$	−5.09121	−	30.9411i	−0.170562	−	1.03656i
$892$	−5.19193			−0.173839
$893$	−20.4113			−0.683037
$894$	−11.4024	−	4.18714i	−0.381354	−	0.140039i
$895$			12.4614i			0.416539i
$896$	3.80421			0.127090
$897$	−26.1089	−	4.95144i	−0.871749	−	0.165324i
$898$	8.98085			0.299695
$899$			27.9308i			0.931545i
$900$	2.28706	+	1.94149i	0.0762352	+	0.0647162i
$901$	33.6758			1.12190
$902$	13.9038			0.462947
$903$	15.8112	−	43.0570i	0.526163	−	1.43285i
$904$			9.34208i			0.310713i
$905$		−	10.2922i		−	0.342124i
$906$	23.2752	+	8.54699i	0.773265	+	0.283955i
$907$		−	23.3635i		−	0.775772i	−0.921707	−	0.387886i	$-0.873206\pi$
$907$		−	23.3635i		−	0.775772i	0.921707	−	0.387886i	$-0.126794\pi$
$908$	−16.7298			−0.555199
$909$	17.7748	−	20.9385i	0.589551	−	0.694487i
$910$	12.1703			0.403440
$911$	−54.0556			−1.79094			−0.895471	−	0.445119i	$-0.853161\pi$
$911$	−54.0556			−1.79094			−0.895471	+	0.445119i	$0.853161\pi$
$912$	3.06151	+	1.12423i	0.101377	+	0.0372271i
$913$	44.6275			1.47695
$914$	5.68472			0.188034
$915$	−5.39692	−	1.98183i	−0.178417	−	0.0655173i
$916$		−	18.2985i		−	0.604599i
$917$	−40.1357			−1.32540
$918$	−12.1838	−	21.5278i	−0.402126	−	0.710523i
$919$		−	26.4439i		−	0.872303i	−0.899873	−	0.436152i	$-0.856341\pi$
$919$		−	26.4439i		−	0.872303i	0.899873	−	0.436152i	$-0.143659\pi$
$920$	−4.73108	+	0.785398i	−0.155979	+	0.0258938i
$921$	−11.5813	+	31.5382i	−0.381617	+	1.03922i
$922$	−23.4737			−0.773066
$923$			31.7537i			1.04519i
$924$	−21.5501	−	7.91354i	−0.708948	−	0.260336i
$925$		−	4.88527i		−	0.160627i
$926$		−	41.4142i		−	1.36095i
$927$	31.1791	−	36.7288i	1.02406	−	1.20633i
$928$	7.50582			0.246391
$929$		−	0.525207i		−	0.0172315i	−0.999963	−	0.00861575i	$-0.997257\pi$
$929$		−	0.525207i		−	0.0172315i	0.999963	−	0.00861575i	$-0.00274251\pi$
$930$	−6.05031	−	2.22176i	−0.198397	−	0.0728545i
$931$			14.0696i			0.461111i
$932$		−	6.92267i		−	0.226759i
$933$	27.4434	+	10.0776i	0.898456	+	0.329927i
$934$			28.5266i			0.933420i
$935$	−16.5863			−0.542430
$936$	−6.21113	+	7.31666i	−0.203017	+	0.239153i
$937$		−	22.6417i		−	0.739673i	−0.929097	−	0.369836i	$-0.879414\pi$
$937$		−	22.6417i		−	0.739673i	0.929097	−	0.369836i	$-0.120586\pi$
$938$		−	14.0923i		−	0.460130i
$939$	43.4530	+	15.9566i	1.41803	+	0.520724i
$940$		−	10.8399i		−	0.353559i
$941$	22.1089			0.720731			0.360366	−	0.932811i	$-0.382652\pi$
$941$	22.1089			0.720731			0.360366	+	0.932811i	$0.382652\pi$
$942$	9.78475	−	26.6458i	0.318804	−	0.868168i
$943$	−18.8800	+	3.13423i	−0.614816	+	0.102064i
$944$			7.35646i			0.239432i
$945$	−9.73626	−	17.2032i	−0.316721	−	0.559619i
$946$	24.2539			0.788562
$947$		−	37.4498i		−	1.21696i	−0.793571	−	0.608478i	$-0.791780\pi$
$947$		−	37.4498i		−	1.21696i	0.793571	−	0.608478i	$-0.208220\pi$
$948$	−16.4644	−	6.04597i	−0.534739	−	0.196364i
$949$	10.1885			0.330731
$950$	1.88297			0.0610917
$951$	−12.2780	−	4.50868i	−0.398142	−	0.146204i
$952$	−18.1100			−0.586950
$953$	39.1208			1.26725			0.633624	−	0.773641i	$-0.281567\pi$
$953$	39.1208			1.26725			0.633624	+	0.773641i	$0.281567\pi$
$954$	−13.7340	+	16.1785i	−0.444654	+	0.523800i
$955$	−22.5181			−0.728668
$956$		−	2.10045i		−	0.0679335i
$957$	−42.5191	−	15.6137i	−1.37445	−	0.504718i
$958$			3.94290i			0.127389i
$959$		−	61.8617i		−	1.99762i
$960$	−0.597052	+	1.62589i	−0.0192698	+	0.0524755i
$961$	−17.1525			−0.553307
$962$	15.6288			0.503892
$963$	−40.2967	−	34.2080i	−1.29854	−	1.10234i
$964$			0.0258170i			0.000831511i
$965$	−12.3106			−0.396291
$966$	31.0467	+	5.88789i	0.998912	+	0.189440i
$967$	−37.1788			−1.19559			−0.597795	−	0.801649i	$-0.703956\pi$
$967$	−37.1788			−1.19559			−0.597795	+	0.801649i	$0.703956\pi$
$968$		−	1.13914i		−	0.0366134i
$969$	−14.5744	−	5.35195i	−0.468198	−	0.171929i
$970$	−2.98155			−0.0957317
$971$	−27.4914			−0.882240			−0.441120	−	0.897448i	$-0.645419\pi$
$971$	−27.4914			−0.882240			−0.441120	+	0.897448i	$0.645419\pi$
$972$	15.3113	+	2.92632i	0.491111	+	0.0938618i
$973$		−	88.8034i		−	2.84691i
$974$			1.02454i			0.0328283i
$975$	−1.91007	+	5.20149i	−0.0611711	+	0.166581i
$976$		−	3.31936i		−	0.106250i
$977$	−41.7887			−1.33694			−0.668469	−	0.743740i	$-0.733050\pi$
$977$	−41.7887			−1.33694			−0.668469	+	0.743740i	$0.733050\pi$
$978$	−8.56074	−	3.14363i	−0.273742	−	0.100522i
$979$	38.5030			1.23056
$980$	−7.47199			−0.238684
$981$	24.7182	−	29.1179i	0.789192	−	0.929662i
$982$	12.3413			0.393826
$983$	−58.4135			−1.86310			−0.931550	−	0.363612i	$-0.881543\pi$
$983$	−58.4135			−1.86310			−0.931550	+	0.363612i	$0.881543\pi$
$984$	−2.38261	+	6.48833i	−0.0759549	+	0.206840i
$985$		−	0.321151i		−	0.0102327i
$986$	−35.7317			−1.13793
$987$	−24.6208	+	67.0473i	−0.783688	+	2.13414i
$988$			6.02393i			0.191647i
$989$	−32.9343	+	5.46735i	−1.04725	+	0.173852i
$990$	6.76439	−	7.96840i	0.214986	−	0.253252i
$991$	55.2646			1.75554			0.877768	−	0.479085i	$-0.159031\pi$
$991$	55.2646			1.75554			0.877768	+	0.479085i	$0.159031\pi$
$992$		−	3.72122i		−	0.118149i
$993$	−15.1338	+	41.2124i	−0.480257	+	1.30783i
$994$		−	37.7592i		−	1.19765i
$995$			14.6664i			0.464958i
$996$	−7.64752	+	20.8257i	−0.242321	+	0.659889i
$997$	−45.4910			−1.44071			−0.720357	−	0.693604i	$-0.756022\pi$
$997$	−45.4910			−1.44071			−0.720357	+	0.693604i	$0.756022\pi$
$998$		−	29.3987i		−	0.930599i
$999$	−12.5031	−	22.0919i	−0.395580	−	0.698956i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.e.a.551.12	yes	16
3.2	odd	2		690.2.e.b.551.4	yes	16
23.22	odd	2		690.2.e.b.551.12	yes	16
69.68	even	2	inner	690.2.e.a.551.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.e.a.551.4	✓	16	69.68	even	2	inner
690.2.e.a.551.12	yes	16	1.1	even	1	trivial
690.2.e.b.551.4	yes	16	3.2	odd	2
690.2.e.b.551.12	yes	16	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(-0.597052 + 1.62589i) q^{3} -1.00000 q^{4} -1.00000 q^{5} +(-1.62589 - 0.597052i) q^{6} +3.80421i q^{7} -1.00000i q^{8} +(-2.28706 - 1.94149i) q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +(-0.597052 + 1.62589i) q^{3} -1.00000 q^{4} -1.00000 q^{5} +(-1.62589 - 0.597052i) q^{6} +3.80421i q^{7} -1.00000i q^{8} +(-2.28706 - 1.94149i) q^{9} -1.00000i q^{10} -3.48413 q^{11} +(0.597052 - 1.62589i) q^{12} +3.19916 q^{13} -3.80421 q^{14} +(0.597052 - 1.62589i) q^{15} +1.00000 q^{16} -4.76053 q^{17} +(1.94149 - 2.28706i) q^{18} -1.88297i q^{19} +1.00000 q^{20} +(-6.18523 - 2.27131i) q^{21} -3.48413i q^{22} +(0.785398 + 4.73108i) q^{23} +(1.62589 + 0.597052i) q^{24} +1.00000 q^{25} +3.19916i q^{26} +(4.52214 - 2.55934i) q^{27} -3.80421i q^{28} -7.50582i q^{29} +(1.62589 + 0.597052i) q^{30} -3.72122 q^{31} +1.00000i q^{32} +(2.08021 - 5.66482i) q^{33} -4.76053i q^{34} -3.80421i q^{35} +(2.28706 + 1.94149i) q^{36} -4.88527i q^{37} +1.88297 q^{38} +(-1.91007 + 5.20149i) q^{39} +1.00000i q^{40} +3.99062i q^{41} +(2.27131 - 6.18523i) q^{42} +6.96125i q^{43} +3.48413 q^{44} +(2.28706 + 1.94149i) q^{45} +(-4.73108 + 0.785398i) q^{46} -10.8399i q^{47} +(-0.597052 + 1.62589i) q^{48} -7.47199 q^{49} +1.00000i q^{50} +(2.84229 - 7.74011i) q^{51} -3.19916 q^{52} -7.07396 q^{53} +(2.55934 + 4.52214i) q^{54} +3.48413 q^{55} +3.80421 q^{56} +(3.06151 + 1.12423i) q^{57} +7.50582 q^{58} +7.35646i q^{59} +(-0.597052 + 1.62589i) q^{60} -3.31936i q^{61} -3.72122i q^{62} +(7.38582 - 8.70044i) q^{63} -1.00000 q^{64} -3.19916 q^{65} +(5.66482 + 2.08021i) q^{66} +3.70440i q^{67} +4.76053 q^{68} +(-8.16116 - 1.54773i) q^{69} +3.80421 q^{70} +9.92565i q^{71} +(-1.94149 + 2.28706i) q^{72} +3.18473 q^{73} +4.88527 q^{74} +(-0.597052 + 1.62589i) q^{75} +1.88297i q^{76} -13.2543i q^{77} +(-5.20149 - 1.91007i) q^{78} -10.1264i q^{79} -1.00000 q^{80} +(1.46126 + 8.88058i) q^{81} -3.99062 q^{82} -12.8088 q^{83} +(6.18523 + 2.27131i) q^{84} +4.76053 q^{85} -6.96125 q^{86} +(12.2037 + 4.48137i) q^{87} +3.48413i q^{88} -11.0510 q^{89} +(-1.94149 + 2.28706i) q^{90} +12.1703i q^{91} +(-0.785398 - 4.73108i) q^{92} +(2.22176 - 6.05031i) q^{93} +10.8399 q^{94} +1.88297i q^{95} +(-1.62589 - 0.597052i) q^{96} -2.98155i q^{97} -7.47199i q^{98} +(7.96840 + 6.76439i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 16 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) 16 * q - 16 * q^4 - 16 * q^5 + 2 * q^6 + 2 * q^9	\( 16 q - 16 q^{4} - 16 q^{5} + 2 q^{6} + 2 q^{9} - 12 q^{11} - 12 q^{14} + 16 q^{16} + 8 q^{18} + 16 q^{20} - 4 q^{21} + 4 q^{23} - 2 q^{24} + 16 q^{25} + 24 q^{27} - 2 q^{30} + 4 q^{31} - 28 q^{33} - 2 q^{36} - 16 q^{38} - 8 q^{39} + 12 q^{44} - 2 q^{45} - 4 q^{46} - 4 q^{49} - 2 q^{51} - 8 q^{53} - 26 q^{54} + 12 q^{55} + 12 q^{56} + 28 q^{57} - 8 q^{58} - 16 q^{64} + 10 q^{66} + 30 q^{69} + 12 q^{70} - 8 q^{72} - 16 q^{73} - 24 q^{74} - 12 q^{78} - 16 q^{80} + 22 q^{81} - 16 q^{82} - 40 q^{83} + 4 q^{84} - 40 q^{86} + 20 q^{87} + 80 q^{89} - 8 q^{90} - 4 q^{92} - 4 q^{93} - 24 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100}) \) 16 * q - 16 * q^4 - 16 * q^5 + 2 * q^6 + 2 * q^9 - 12 * q^11 - 12 * q^14 + 16 * q^16 + 8 * q^18 + 16 * q^20 - 4 * q^21 + 4 * q^23 - 2 * q^24 + 16 * q^25 + 24 * q^27 - 2 * q^30 + 4 * q^31 - 28 * q^33 - 2 * q^36 - 16 * q^38 - 8 * q^39 + 12 * q^44 - 2 * q^45 - 4 * q^46 - 4 * q^49 - 2 * q^51 - 8 * q^53 - 26 * q^54 + 12 * q^55 + 12 * q^56 + 28 * q^57 - 8 * q^58 - 16 * q^64 + 10 * q^66 + 30 * q^69 + 12 * q^70 - 8 * q^72 - 16 * q^73 - 24 * q^74 - 12 * q^78 - 16 * q^80 + 22 * q^81 - 16 * q^82 - 40 * q^83 + 4 * q^84 - 40 * q^86 + 20 * q^87 + 80 * q^89 - 8 * q^90 - 4 * q^92 - 4 * q^93 - 24 * q^94 + 2 * q^96 - 8 * q^99

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